What Are Numbers?

Stage: 2, 3, 4 and 5

Article by Toni Beardon

The Question

What are numbers? Most people, even five year olds, can answer
that question to their own satisfaction. Many different answers are
given to this question, all more or less acceptable within the
discourse taking place. Although negative, fractional and
irrational numbers were accepted by scholars in Europe in the
sixteenth century, and earlier in ancient civilisations in other
parts of the world, until the nineteenth century negative numbers
and complex numbers were often disparagingly referred to as absurd
numbers and imaginary numbers. These numbers now play an essential
part in mathematics, even school mathematics, and schoolchildren
learn about them.

This brief descriptive article is intended as light reading
for the general reader. Here we shall explore the question in a
very informal way. We shall discuss different sets of numbers,
including quaternions and a brief mention of Clifford Algebras,
starting with counting numbers and meeting new sets of numbers,
each set containing within it the set of numbers discussed so far.
Quaternions are explored in more detail in the NRICH problem
Two and Four Dimensional Numbers

In order to understand why there are different sorts of
numbers, and what they are, we need to consider how young people
meet the familiar number systems and how we broaden our ideas of
number as we learn more arithmetic. Many small children are proud
of themselves when they can count to one hundred and a little later
they have an experience of awe and wonder when they first
appreciate that counting goes on for ever with no end. These
children have a familiarity with counting and the natural numbers,
even with the concept of infinity, before they meet a number
system. Roughly speaking a number system is a set of entities that
can be combined, according to agreed rules, by the operations of
addition, subtraction, multiplication and division, always
producing answers that are in the set.

Rules of
Arithmetic

Adding and sharing are transactions that we engage in early in
our lives and they take us beyond simple counting into the realm of
arithmetic. So we should define numbers to be more than labels for
naming and recording the size of collections of objects that we
have counted. We need to think of numbers as entities that can be
combined according to an agreed set of rules which we call
arithmetic. The more we know about and use this arithmetic, the
more we appreciate that it can be generalised and refined into more
and more useful mathematical tools for solving human problems of
all sorts.

If we use numbers to describe the size of a collection of
objects we need a number for a set that has nothing in it. So we
need to expand our concept of number to include zero. While the use
of numbers, including place value, dates back at least five
thousand years, scholars in Europe were still debating whether zero
could be a number as recently as five hundred years ago.

Inverses

Once we have an operation which combines two numbers to give
another number, can we undo that process? If we can add 7 what is
the operation on the answer which restores it to the original
number?

Subtraction is another natural idea based on concrete
experience, not of increasing the size of a collection of objects
by combining two collections, but of reducing the size of a
collection by removing some of the objects. If we have $5$ coins
and we need $12$ to buy something we can ask how many more coins do
we need, which number do we add to five to make $12$; this is
equivalent to finding the answer to $12 - 5$, but what number do we
add to $13$ to make $9$ or what is the answer to $9 - 13$? If we
say that there is no answer to such subtractions then it is only
because we do not know about negative numbers. Once negative
numbers come onto the scene we have many uses for them.

We can do arithmetic without knowing the mathematical language
and nothing so far is beyond the experience of a small child
learning to read a thermometer on a wintry day.

We think of any subtraction as simply the addition of two
integers so subtraction is not essential to recording arithmetic
operations. For example $9 - 13 = 9 + (-13) = -4$. The integer
$-13$ is called the inverse of the $+13$ because $(+13) + (-13) =
0$. Every integer has an inverse such that the number added to its
inverse gives zero. We are already working with the structure and
using the rules which define the arithmetic involved in adding
integers. This is an example of a mathematical structure called a
group .

From the Natural Numbers to the Integers

Another way to explain the evolution of thinking that extends
ideas of counting and addition is to say that if we want to be able
to solve all equations of the form $a + x = b$, where $a$ and $b$
are given and we have to find $x$, then for all such equations to
have solutions we need to work in the set of integers. For example
$9 + x = 5$ had no solution within the set of counting numbers.

Rational Numbers

In the same way as horizons are extended to include negative
numbers it is also everyone's experience to learn that fractions
are also numbers. Mathematicians call these numbers rational
numbers . Children are interested in 'fair shares' even before
they start school so division is also based on concrete
experience.

It is not until we can work with the set of rational numbers
that every addition, subtraction, multiplication and division of
two numbers in the set gives an answer that is also a number in the
set giving a 'self contained' number system. The rules for the
arithmetic of rational numbers are simple. This set of rules
defines what mathematicians call a field.

(1) The set of rational numbers is closed under addition, and
associative, the rational number zero is the additive identity and
every rational number has an additive inverse. We say rational
numbers form a commutative group under addition.

(2) The set of rational numbers, leaving out the number zero, is
closed under multiplication, and associative, the rational number
one is the multiplicative identity and every rational number in
this set has a multiplictive inverse. We say the rational numbers,
leaving out the number zero, form a commutative group under
multiplication.

(3) When we add and multiply rational numbers we use the
distributive law. For example $$3\times (4 + 5) = 3 \times 4 + 3
\times 5 = 27.$$ We need to extend our ideas of the arithmetic of
whole numbers to include fractions (rational numbers) because we
cannot solve all equations of the form $ax = b$, where $a$, $b$ and
$x$ are integers, $a$ and $b$ are given and we have to find $x$. If
$a, b$ and $x$ are rational numbers then all such equations have
solutions.

Irrational Numbers

Many people use only rational numbers because, even though no
rational number will give exact measurements of even simple shapes,
exact measurements can be approximated to a high degree of accuracy
by rational numbers. The length of the diagonal of a unit square is
$\sqrt 2$ and this is an irrational number, one that cannot be
written as the quotient of two integers. It is approximately
$1.414$ but it cannot be given exactly however many decimal places
we use. See the interactive proof sorter for the proof that

The rational and irrational numbers together make up the
real numbers. Each real number corresponds to exactly one
point on a line and all the points on that line are represented by
real numbers. We call this line the real line . The real
numbers are equivalent to one dimensional vectors and, together
with addition and multiplication, form a field. We have now
discussed two different examples of fields of numbers, the
rationals and the reals.

Complex Numbers

In the field of real numbers we can solve the equation $x^2 = a$
only when $a$ is positive but not when $a$ is negative. Real
numbers are good enough for many mathematical purposes but clearly
they have limitations. It is necessary to recognise the existence
of two dimensional numbers, the complex numbers, in order to solve
all quadratic equations. See
Root Tracker and
Cubic Tracker.

In 1799 Gauss proved the Fundamental Theorem of Algebra: every
polynomial equation over the complex numbers has a full set of
complex solutions. Then, and finally, complex numbers were
completely accepted as 'proper' numbers. This theorem means that
every quadratic equation has two solutions, every cubic has three
and so on.

What else can we do with complex numbers? Complex numbers are
two dimensional; whereas real numbers correspond to points on a
line, complex numbers correspond to points in the plane. The
complex number written as $x+yi$ corresponds to the point in the
plane with coordinates $(x,y)$. Let us examine the significance of
this mysterious $i$ referred to as an 'imaginary' number. What role
does it play?

Complex Numbers and Rotations

Think about taking a real number and finding its additive
inverse, say $5$ and $-5$. To move from any real number to its
additive inverse we must multiply by $-1$, or to think of it
another way, we must move from the positive real axis to the
negative real axis, a half turn about the origin so the point $(5,
0)$ moves to $(-5,0)$, that is the complex number $5+ [0\times i]$
moves to $-5+[0\times i]$. No obvious clue there as to the role of
$i$ but let's probe a bit further and think more about rotations.
Two quarter turns make a half turn so what happens when we rotate
the plane by a quarter turn about the origin? The point $(5, 0)$
moves to $(0,5)$, that is the complex number $5 + [0\times i]$
moves to the complex number $0 + 5 i$ which appears to be
equivalent to multiplying by $i$, that is $i(5 + [0\times i] ) = 0
+ 5i$. (It is not necessary to write in $[0\times i]$ here but we
do so to make clear how the mappings of the complex numbers
correspond to the mappings of the points in the plane including the
points on the real line which correspond to real numbers.)

We have seen that a real number is mapped to its additive
inverse by multiplying by $-1$. So if a quarter turn of the complex
plane is equivalent to multiplying by i then a half turn (that is
two quarter turns) must be equivalent to multiplying by $i$ twice
which must be the same as multiplying by -1 and this tells us that
$i^2 = -1$. Taking $i^2 = -1$ this fits in with moving $(5, 0)$ to
$(-5, 0)$, or correspondingly, $5 + [0\times i]$ to $i^2(5 +
[0\times i]) = -5 + [0\times i]$. A quarter turn moves $(0,5)$ to
$(-5,0)$ or equivalently $0 + 5i$ to $-5+ [0\times i]$ and this
time multiplying by $i$ gives $i(0 + 5 i) = 5i^2 + [0\times i] = -5
+ [0\times i]$.

All this works beautifully because $i^2 = -1$. This little
complex number, corresponding to the point $(0,1)$, not only allows
all polynomial equations to have solutions but gives a powerful
tool for working with rotations. In the form $x+yi$ complex numbers
can be added, subtracted, multiplied and divided according the the
same rules as elementary arithmetic, that is the complex numbers
form a field. We now have three fields of numbers, the complex
numbers, the reals and the rationals.

Another glimpse of the beauty of complex numbers is seen in the
formula $$e^{i\theta}=\cos \theta + i\sin \theta $$ linking
geometry, trigonometry and analysis. This formula involves the
important real number $e$ as well as the complex number $i$.
Students usually meet this formula and use it in their last year in
school if they are preparing to study mathematics, physics or
engineering in higher education. In this formula the angle $\theta$
is given in radians and not in degrees but the conversion is a
simple matter because $\pi$ radians is $180^o$. If we put $\theta =
\pi$ we have the very beautiful result $$e^{i\pi} = \cos \pi +
i\sin \pi$$ which connects the important numbers $e$, $i$, $\pi$
and -1 in the simple little formula $$e^{i\pi} = -1.$$ If we put
$\theta = {\pi \over 2}$ we have $$e^{i{\pi \over 2}} = \cos {\pi
\over 2} + i\sin {\pi \over 2} = i$$ which suggests that
multiplying by $$e^{i\theta}=\cos \theta + i\sin \theta $$ might be
equivalent to rotating the complex plane by an angle $\theta$ ( as
this works for quarter turns and $i$) and this is indeed the
case.

If one dimensional real numbers can be generalised to two
dimensional complex numbers and both systems form fields, the
obvious question is "what about higher dimensional numbers?"

Three dimensional numbers do not exist

Three dimensional vectors are of fundamental importance in
applied mathematics. They can be added and subtracted but although
there are two different types of vector multiplication,
multiplicative inverses do not exist and so the set of three
dimensional vectors do not form a field and cannot be a set of
numbers. To understand the significance of the two alternative
definitions of vector multiplication it is necessary to know about
four dimensional quaternions.

Quaternions

If we think about rotations of the plane, and $i$ as the key to
understanding the essence of complex numbers, what about rotations
of 3 dimensional space? For rotations of the plane that map the
plane to itself there is only one possible axis of rotation which
must be perpendicular to the plane. [Without loss of generality we
can take the centre of rotation to be at the origin.] However in 3
dimensional space there are infinitely many possible axes of
rotation through the origin. If we want to specify an axis of
rotation we need the three coordinates for one other point on the
axis. Whereas in the complex plane we only need one parameter to
specify a rotation (the angle of rotation), in 3 dimensional space
we need four parameters (three to specify the axis of rotation and
one to specify the angle of rotation). This takes us to four
dimensions and explains why there are two and four dimensional
numbers, but not three dimensional numbers, and why quaternions
provide a very efficient way to work with rotations of 3
dimensional space.

Quaternions, discovered by the Irish mathematician Sir William
Rowan Hamilton in 1843, have all the properties of a field except
that multiplication is not commutative. Moreover quaternions
incorporate three dimensional vectors and much of vector algebra
and provide simple equations for reflections and rotations in three
dimensional space.

As applied mathematics and physics regularly deal with motion in
space, quaternions are very useful. It is a quirk of history that
this was not perhaps fully appreciated at first and vector algebra
was invented as a tool to work with motion in 3 dimensional space
and concentrate attention on only 3 dimensions. However a lot of
the simplicity of the equations involving quaternions was lost as
well as sight of the underlying reasons for defining scalar and
vector multiplication in the way they are defined. Nowadays
quaternions have come into their own again as an important tool
frequently used in applied mathematics and theoretical physics.
Quaternions are also now widely used in programming computer
graphics because the quaternion algebra involved in transformations
in 3 dimensions is so simple.

What about higher dimensional numbers? Number theorists work
with Clifford Algebras, named after William Clifford (1845-1879),
which generalise complex numbers and quaternions to dimensions 2,
4, 8, 16... and higher dimensions (all powers of 2). Some of the
properties of a field are lost, for example quaternions are not
commutative under multiplication, but Clifford algebras are
associative. Clifford algebras have important applications in a
variety of areas including geometry and theoretical physics. Other
generalisations are studied for which multiplication is not
associative.

This 'big picture' discourse has ranged, without getting too
technical, from kindergarten mathematics to the fringe of research
into analysis and applications of number. There is a wealth of
literature to take the reader further at every level and the links
below may provide a useful start on such a journey of
discovery.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.